/m/bill_james

Reader Comments and Retorts

Statements posted here are those of our readers and do not represent the BaseballThinkFactory. Names are provided by the poster and are not verified. We ask that posters follow our submission policy. Please report any inappropriate comments.

The usual rule is that a player is consistent when he is young; when he gets older, what he loses is not the ability to produce but the consistency of his production.” Can I take that as fact? Could you, please, elaborate on that? It would make a good subject for a serious study.

I absolutely disagree with that assertion...It is very counter intuitive. If he is saying young means 26-30...then sure I'll agree there, but until someone produces a study on consistency, it is more obvious that young players are inconsistent. (again young is under 26 years old)

1991 she correctly solved an apparently simple question ++ Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

It's simple. When you first choose, you have a 1/3 chance of picking the right door, and a 2/3 chance of picking the wrong door.

The host then eliminates *one of the wrong doors*. If you originally chose incorrectly, which you would have done 2/3 of the time, then you always win by switching. If you originally chose the correct door, which you would have done only 1/3 of the time, you always lose by switching.

Therefore, if you don't switch, you win 1/3 of the time. If you do switch, you win 2/3 of the time.

It's not obvious to me at all, what I think is that very inconsistent young players tend to become more consistent over time- very consistent young players become less consistent over time- why? 1 part regression to the mean, 1 part simple change-

The young consistent player will get banged up as he gets older, he'll heal slower so that physically he will be different from one week to another, one month/year to another

the inconsistent player likely does not have uniform approach, he'll fix things whether they need to be fixed or not, his effort or lack thereof will be inconsistent, his attention span will wander, that type of thing will tamp down over time.

Of course I don't know either-
the thing is James used to throw these things out, he'd reason himself into something- but then he'd TEST it, he'd devise a study, and if he as wrong he'd say so in the next year's Abstract.
A great one was a blurb he wrote on Lloyd Moseby after Moseby's seeming break out year in 1983- he said that looking at, when a player has a clearly established skill level, not an erratic performer, but one with a clearly established level, and then one year plays at a completely different and higher level- isn't that evidence that the change is real? Seems obvious to me, isn't it obvious to you? It wasn't obvious to me, to me Lloyd Moseby going from hitting .235/.290/.360 every year to .315/.376/.499 was obviously a fluke... 20/20 hindsight, 1983 was to be Lloyd's best year, but 1984-91 he did play much better than he had pre-1983.

I think you're right for hitters, but I thought there were studies that showed pitchers are pretty much what they are from the get go.

I don't think I've ever seen anything like that. Most studies I have seen have shown that pitchers prime seasons is older than hitters, and everything I've seen from personal observation has indicated that young pitchers are massively inconsistent. There just doesn't seem to be many pitchers like Maddux/Clemens who can maintain any high consistency for any length of time, most are like less extreme versions of Saberhagen. Even within the season, I think you'll see more inconsistent performances from younger than you will from established veterans.

When I was young and for about the one day I was into bodybuilding, there was talk about why there were few if any championship bodybuilders under 25 years old, and the argument was that young bodies are very elastic from a day to day basis, and it's hard to maintain consistent mass/tone because of youth. Extending that out to athletics, and it can mean your muscles change more rapidly on a day to day basis affecting the consistency. This always made sense to me and I don't think I have ever seen anything contradicting this.

The Monte Hall problem is a simple example of symmetry breaking. Because the host only shows doors with no prize behind them, he's also telling you something about the door he didn't open -- that, for whatever reason, he *didn't* choose it as an empty door.

A simple illustration is to consider the problem with 1000 doors. You pick a door, Monte Hall opens 998 doors and asks if you want to switch. 999 times out of 1000, your initial guess was wrong and you should switch. 1 time out of 1000, your initial guess was right, and you lose out on the prize by switching. The 3 door problem is exactly the same, except the probabilities are 2/3 and 1/3.

Most people get it wrong because they think it's a 50/50 proposition with the second choice, because one of them wins and one of them loses -- they forget that 2/3 of the time they were wrong with their first choice, and Monty opening a losing door doesn't change that -- because he was always going to open a losing door.

I don't think I've ever seen anything like that. Most studies I have seen have shown that pitchers prime seasons is older than hitters, and everything I've seen from personal observation has indicated that young pitchers are massively inconsistent.

Maybe I was thinking of this which says pitchers peak in strikeout rates at 24, but that doesn't necessarily speak to overall performance and consistency.

That study does have prime for pitchers at 29 and hitters at 30, but as you pointed out, different skills(like strikeouts) peak at different times. I am not convinced by that study entirely... I'm sure others have debated it extensively, but by relying on players with longer careers, it's weeding out those who faded quickly thereby making the average higher than it probably is(not by a lot though) (and of course the study completely ignores defense, and baserunning--except 2b+3b--- which means by it's very nature it's weighted to offensive skills that age better)

I've always thought that a large component of age-related decline is actually injury-related decline. I call it the Joe Randa effect -- a player in his 30s who doesn't get injured can stick around a lot longer than you might think.

The Monte Hall problem is a simple example of symmetry breaking. Because the host only shows doors with no prize behind them, he's also telling you something about the door he didn't open -- that, for whatever reason, he *didn't* choose it as an empty door.

Why does it matter? This doesn't change the odds of the second choice at all. It's still a 50/50 proposition.

In fact, because Monty was always going to show an empty door, and because the contestant always gets the second chance, it's wrong to think about the first choice as being a 1/3 proposition. Really the first choice is wholly irrelevant. There's no logic as to why a contestant should automatically change their guess - it's going to be 50/50 either way. When it comes down to it, this isn't really a game about picking among three doors, it's a game about picking among two doors.

Why does it matter? This doesn't change the odds of the second choice at all. It's still a 50/50 proposition.

In fact, because Monty was always going to show an empty door, and because the contestant always gets the second chance, it's wrong to think about the first choice as being a 1/3 proposition. Really the first choice is wholly irrelevant. There's no logic as to why a contestant should automatically change their guess - it's going to be 50/50 either way. When it comes down to it, this isn't really a game about picking among three doors, it's a game about picking among two doors.

Yep, that's exactly the incorrect logic that fools people.

Try it yourself. Use a deck of cards, 52 doors, try to find the Ace of Spades. Then have somebody else (who can look at them) "open" 50 of them that aren't the Ace of Spades and see if you think you should switch then...

Why does it matter? This doesn't change the odds of the second choice at all. It's still a 50/50 proposition.

No, it's a 2/3 : 1/3 proposition.

If the initial guess was correct, switching loses the prize. This happens 1/3 of the time.

If the initial guess was incorrect, the chosen door is empty. Monte Hall shows you the other empty door, which means that switching wins the game. This happens 2/3 of the time.

Conclusion: it's better to switch.

The broken symmetry comes from the fact that Monte only shows you an empty door which you did not initially choose. Thus, he is giving you information about the door you didn't choose, but not about the door which you did choose (since he will never open your door to show you whether it is empty or not).

If Monte could open any empty door, including the one you chose, your 50:50 intuition would be right. The trick to the problem is that he is pretending to do that, but is actually doing something else, and that's why so many people get it wrong.

It's a matter of changing your own mindset- on your first choice, you're not trying to pick the car. You're trying to pick a goat, which you have a 66.6% chance of doing. You switch after the first door is opened since you had the "goat odds" on your side when you made your first choice. Whether Monty opens a door or not does not change your odds that you picked a goat door with your first choice...

If you switch and you're wrong, look on the bright side- the taxes on winning a goat are a lot less than those of winning a car.

Scenario 2: Host randomly chooses a door to open. 1/3 of the time you lose before you get the option to switch. If you switch, you win 1/3 of the time (or 1/2 the time conditional on a losing door being opened). This scenario operates exactly how most people think the problem does. Scenario 2 is Deal or No Deal, with the contestant randomly choosing, by the way.

Scenario 3: Host only opens a losing door when you've chosen correctly, hoping to sucker you into switching. If you switch, you lose 100% of the time.

The original question:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

There is absolutely nothing in the question that says the host is required to open a losing door (scenario 1). Vos Savant's answer, with added emphasis:

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

So she poses the correct scenario where switching makes sense but there's nothing in the original scenario that justifies her assumption. The situation you are in is that you have chosen one door and the host has opened an empty door. Short of other external information, this hasn't changed one damn thing. You have to know what rules the host is operating under.

The likelihood of each scenario is unknown. (It could of course be "known" if in the past 1000 shows, the host has opened a losing door and offered a switch.) If the scenarios are equally likely then the odds are 1/3 if you stick and 1/3 if you switch. The only way it makes sense to switch is if Scenario 1 is more likely than Scenario 3. If 3 is more likely than 1, you should stick. Given a game show host is trying to maintain/increase ratings and assuming audiences like to see contestants win fairly frequently, it might make sense to assume that 1 is more likely than 3. However, if it's some game run by a casino or the state lottery, you can assume they want you to lose more than win (therefore depends on the payout odds). If if it's a street 3-card Monte hustler, 3 seems more likely than 1 (of course the real scenario there is that he palmed the red queen).

By game theory, the host should mix things up a bit to keep contestants guessing but, if his goal is for you to lose more often than win (while maintaining interst), his optimal move is certainly not 100% Scenario 1.

So in essence, the reflexive 50/50 crowd was wrong and vos Savant cheated. If the only information you have is that the host opened an empty door and you have NO information about the host's behavior, then it doesn't matter whether you switch or not.

The various knowledges involved here remind me of the similar Two Children problem (I have 2 kids, one's a boy, what are the odds the other's a boy?) It depends on what one knows about how my family got selected for this puzzle in the first place.

I was one of the few who got the answer right the first time Marilyn Vos Savant (Walt is right there; her birth name was Marilyn Mach, back when she was in Kirkwood, MO, but she changed it to Vos Savant). However, I don't think that was due to any great logic genius on my part; I just recognized the Principle of Restricted Choice when I saw it. I was not sophisticated enough, for example, to realize that the syntax was dicey, and within that logic, it would be possible for the host to open the Car door, instead of only a Goat door, a choice that would remove the restriction. In fact, I'd never given that option a thought until Walt deconstructed the problem here for me. Thanks, Walt. - Brock Hanke

No, you should switch whether or not the host is forced to show you a goat. For any particular instance of making that choice (switch or don't) it matters not one whit what the host has done in the past or why the host is doing what he's doing.

It feels like the host is giving you no information, because you knew that one of the unchosen doors, at least, was a goat. That's what makes the problem so wonderful. You can argue all you want, but it works to switch. Explain it to yourself however you want. It works.

Really the first choice is wholly irrelevant. There's no logic as to why a contestant should automatically change their guess - it's going to be 50/50 either way. When it comes down to it, this isn't really a game about picking among three doors, it's a game about picking among two doors.

It's about picking {one door} vs. picking {the better of the other two doors}. The host is never showing you the better of the other two doors, which means the one he is offering you to switch to is the better of the other two doors. If you think about it precisely that way - that he is offering you the better of doors 2 and 3 - then it's obvious you should switch.

To drive this home, let's say the problem went more strictly as I put it. You choose Door #1. The host, without revealing anything, then offers you the chance to trade Door #1 to get Door #2 plus Door #3 minus a goat. (That is equivalent to "the better of the two doors".) Do you switch? Hell yes. There's a 2/3 chance that works out to a car, and a 1/3 chance it works out to a goat.

Now let's say before the switch offer the host says, "What if I told you at least one of those two doors has a goat? Do you want to keep Door #1?" Hell no. There's a 100% chance at least one of those two doors has a goat; this was true when you chose Door #1, and it's true if you switch to Door #2 + Door #3 - goat. The host stating that one has a goat tells you nothing. The host testifying under oath that one has a goat tells you nothing. The host opening up one of the doors to reveal a goat tells you nothing.

In the problem as originally stated, that's basically what is going on. You choose Door 1. The host offers you the chance to switch to the better of the other two doors, which he has narrowed down by showing the worse of the two. There's a 2/3 probability the better of Doors 2 and 3 is a car. Switch.

Scenario 2: Host randomly chooses a door to open. 1/3 of the time you lose before you get the option to switch. If you switch, you win 1/3 of the time (or 1/2 the time conditional on a losing door being opened). This scenario operates exactly how most people think the problem does. Scenario 2 is Deal or No Deal, with the contestant randomly choosing, by the way.

The original question is phrased to match what happens on the game show Let's Make A Deal. When this scenario played out on the show, the revealed door is never the car. So, yes, without prior knowledge we don't know that the door being revealed won't be the car. But this is kind of like saying without prior knowledge we don't know that an unopened Coke bottle contains Coke. While that's true on a literal level, we can also say that with prior knowledge we can safely assume Scenario 2 is unlikely. (The same can be said for Scenario 3.)

Yes, Vos Savant presented the problem with incorrect rules and that's where all the confusion comes from. She assumes that Monty always opens one unchosen losing door. In that case the contestant should switch because his original door is 1/3 and the remaining closed door is 2/3.

But that doesn't hold true if Monty can immediately open a chosen losing door to go "ha ha you lost" without offering the chance to switch. So he opens another door only if the original choice wins. In that case the contestant should obviously not switch.

And by all accounts, the real show held Monty to neither of those sets of rules, but let him do whatever he wanted. Then there is no mathematical solution at all but you have to outguess Monty. Like a poker bluff.